K-decomposability of Positive Maps

For any C-algebra A let A denote the set of all positive elements in A. A state on a unital C-algebra A is a linear functional ω : A → C such that ω(a) ≥ 0 for every a ∈ A and ω(I) = 1 where I is the unit of A. By S(A) we will denote the set of all states on A. For any Hilbert space H we denote by B(H) the set of all bounded linear operators on H . A linear map φ : A → B between C-algebras is called positive if φ(A) ⊂ B. For k ∈ N we consider a map φk : Mk(A) → Mk(B) where Mk(A) and Mk(B) are the algebras of k× k matrices with coefficients from A and B respectively, and φk([aij ]) = [φ(aij)]. We say that φ is k-positive if the map φk is positive. The map φ is said to be completely positive when it is k-positive for every k ∈ N. A Jordan morphism between C-algebras A and B is a linear map ρ : A →